Alg April 2, 2009
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The document contains information about solving different types of math problems: 1) It gives the steps to solve the problem of finding an unknown number if half the number equals 4 more than twice the number. 2) It provides an overview of approximating square root values and discusses graphing quadratic functions, including finding the vertex and x- and y-intercepts. 3) It introduces the quadratic formula as another way to solve quadratic equations and gives an example of using the formula.
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Quadratic equations contain a variable that is squared and have the standard form of ax2 + bx + c = 0, where a cannot be 0. Some examples of quadratic equations are 2x2 - 5x + 3 = 0 and x2 - 3x = 0. Quadratic equations are commonly used to calculate rates of acceleration and deceleration, determine optimal selling prices based on costs, and calculate surface areas based on circumferences.
Chapter 1.4 Opposites, Integers & Absolute Value
The document provides an agenda for an October 28 math class. It includes a do now, homework check on integers and absolute value using a Frayer model vocabulary template, and homework assignments from a pre-algebra textbook. Expressions are categorized as variable expressions, numerical expressions, or variables. Ways to write multiplication and division are demonstrated, along with examples of evaluating variable expressions by substituting values for variables. Vocabulary on evaluating, integers, opposite numbers, and absolute value are defined using Frayer models.
Squaring trick(2)
This document provides tricks for squaring one, two, and three digit numbers.
For single digit numbers, the duplex is the square of the number. For two-digit numbers, the duplex is twice the product of the digits. For three-digit numbers, the duplex is the sum of the first and last digit duplex plus the middle digit duplex. When squaring two-digit numbers, the duplexes of the individual digits are written left to right. For three-digit numbers, the duplexes are written left to right with the full number duplex in the middle. An alternative trick is to subtract the number from the next hundred, square the difference and add it to the original number.
Extreme values
The document provides a solution to find the extreme values of the function f(x)= 2x3 − 15x2 + 24x + 7 on the interval [0, 6]. It outlines a two step process of first finding the critical points by solving the derivative, which are 1 and 4. The second step is to calculate the y-values at these critical points and the endpoints and compare them, with the maximum being f(6)=43 and the minimum being f(4)=-9.
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This document provides examples and steps for adding, subtracting, multiplying, and expanding binomial expressions. It demonstrates adding and subtracting polynomials vertically and horizontally. For multiplication, it explains distributing and combining like terms. It also gives the formulas and examples for sum and difference of squares, square of a binomial, and cubing a binomial expression.
Special factor patterns
The document provides examples and rules for factoring different types of expressions:
1) It shows how to factor expressions of the form a^2 - b^2 into (a+b)(a-b) and gives an example.
2) It explains how to factor square trinomials of the form x^2 + bx + c by finding two factors of c that sum to b.
3) It discusses factoring expressions involving squares and how exponents work.
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The document provides the mathematical definition of absolute value as x if x is positive, 0 if x is 0, and -x if x is negative. It then gives examples of absolute value expressions and their meanings in terms of distance on a number line. It shows how to solve absolute value equations by setting the expression equal to both the positive and negative values. Finally, it lists homework problems from page 55 that include absolute value expressions.
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This document discusses quadratic functions and their graphs. It defines a quadratic function as having the form y = ax2 + bx + c. The graph of a quadratic function is a parabola, which has a vertex (lowest or highest point) and an axis of symmetry (vertical line through the vertex). It provides examples of graphing different quadratic functions using a graphing calculator and by hand, including how to find and plot the vertex and axis of symmetry.
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Quadratic equations contain a variable that is squared and have the standard form of ax2 + bx + c = 0, where a cannot be 0. Some examples of quadratic equations are 2x2 - 5x + 3 = 0 and x2 - 3x = 0. Quadratic equations are commonly used to calculate rates of acceleration and deceleration, determine optimal selling prices based on costs, and calculate surface areas based on circumferences.
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The document provides an agenda for an October 28 math class. It includes a do now, homework check on integers and absolute value using a Frayer model vocabulary template, and homework assignments from a pre-algebra textbook. Expressions are categorized as variable expressions, numerical expressions, or variables. Ways to write multiplication and division are demonstrated, along with examples of evaluating variable expressions by substituting values for variables. Vocabulary on evaluating, integers, opposite numbers, and absolute value are defined using Frayer models.
Squaring trick(2)
This document provides tricks for squaring one, two, and three digit numbers.
For single digit numbers, the duplex is the square of the number. For two-digit numbers, the duplex is twice the product of the digits. For three-digit numbers, the duplex is the sum of the first and last digit duplex plus the middle digit duplex. When squaring two-digit numbers, the duplexes of the individual digits are written left to right. For three-digit numbers, the duplexes are written left to right with the full number duplex in the middle. An alternative trick is to subtract the number from the next hundred, square the difference and add it to the original number.
Extreme values
The document provides a solution to find the extreme values of the function f(x)= 2x3 − 15x2 + 24x + 7 on the interval [0, 6]. It outlines a two step process of first finding the critical points by solving the derivative, which are 1 and 4. The second step is to calculate the y-values at these critical points and the endpoints and compare them, with the maximum being f(6)=43 and the minimum being f(4)=-9.
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This document provides examples and steps for adding, subtracting, multiplying, and expanding binomial expressions. It demonstrates adding and subtracting polynomials vertically and horizontally. For multiplication, it explains distributing and combining like terms. It also gives the formulas and examples for sum and difference of squares, square of a binomial, and cubing a binomial expression.
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The document provides examples and rules for factoring different types of expressions:
1) It shows how to factor expressions of the form a^2 - b^2 into (a+b)(a-b) and gives an example.
2) It explains how to factor square trinomials of the form x^2 + bx + c by finding two factors of c that sum to b.
3) It discusses factoring expressions involving squares and how exponents work.
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The document provides the mathematical definition of absolute value as x if x is positive, 0 if x is 0, and -x if x is negative. It then gives examples of absolute value expressions and their meanings in terms of distance on a number line. It shows how to solve absolute value equations by setting the expression equal to both the positive and negative values. Finally, it lists homework problems from page 55 that include absolute value expressions.
5.1graphquadratics
This document discusses quadratic functions and their graphs. It defines a quadratic function as having the form y = ax2 + bx + c. The graph of a quadratic function is a parabola, which has a vertex (lowest or highest point) and an axis of symmetry (vertical line through the vertex). It provides examples of graphing different quadratic functions using a graphing calculator and by hand, including how to find and plot the vertex and axis of symmetry.
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To compare and perform operations on decimals:
- Stack the decimals according to their decimal points and fill in empty spots with zeros
- To compare, look at the digits from left to right
- To add or subtract, also stack the decimals according to their decimal points and fill in empty spots with zeros, then add or subtract the digits and drop the decimal point in the answer
The document provides examples of comparing, ordering, adding, and subtracting decimals.
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This document contains notes and materials for a Math 9 class covering quadratic functions for the 6th week of the 1st quarter. It includes:
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5. Key parts of graphs of quadratic functions including the vertex, axis of symmetry, x- and y-intercepts, and the effect of the leading coefficient on
Inequalities
This document provides information and examples about solving inequalities:
1) It gives examples of solving linear equations with one variable and two variables.
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3) It provides rules for solving inequalities, such as reversing the inequality sign when multiplying or dividing both sides by a negative number.
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The document provides steps for subtracting and adding fractions. It explains that to subtract fractions, you find a common factor, cross multiply the terms, and then subtract them. The difference is then placed over the product of the original denominators. An example problem subtracts 9/5 - 4/3 and gets 7/12. For adding fractions, the same steps are followed except the terms are added instead of subtracted. An example adds 11/3 + 5/4 and gets 53/60.
Edu 290 ppt 2
The document provides steps for subtracting and adding fractions. It explains that to subtract fractions, you find a common factor, cross multiply the terms, and then subtract them. The difference is then divided by the product of the common factors to simplify the fraction. For adding fractions, the same steps are followed but the terms are added instead of subtracted. Two examples demonstrating these steps are provided.
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This document discusses quadratic functions in vertex form. It defines quadratic functions as having an x^2 term as the highest power of x. Vertex form is defined as y = a(x - h)^2 + k, where (h, k) is the vertex. The document shows how to graph functions in vertex form by sketching the parent graph y = x^2 and then performing transformations based on the a, h, and k values in the function. Several examples are worked through to demonstrate finding the vertex and axis of symmetry.
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The document provides examples and steps for factoring different types of polynomials:
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2) It demonstrates factoring a perfect square trinomial by identifying the pattern a^2 + 2ab + b^2 and grouping terms.
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The document provides a date of May 21, 2009. No other information is given in the single line document, which simply states the date with no other context provided. In just one line, the document communicates only the date of May 21, 2009 without any additional details.
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The document provides instructions and notes for students regarding classwork and homework assignments. It includes warm-up questions for students to fill out on index cards and notes on sets of numbers, order of operations, and examples of classwork problems. Students are also assigned to get a Blogger account and explore textbook resources online by specific dates.
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The document contains instructions for students to complete worksheet problems involving adding and subtracting fractions. It lists several fraction addition and subtraction problems for students to work through independently, including problems finding the sum of fractions with common denominators and fractions with algebraic expressions in the denominators. It also instructs students to find the vertex of a parabolic function and explain what the graph of that function will look like.
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The document provides steps for subtracting and adding fractions. It explains that to subtract fractions, you find a common factor, cross multiply the terms, and then subtract them. The difference is then placed over the product of the original denominators. An example problem subtracts 9/5 - 4/3 and gets 7/12. For adding fractions, the same steps are followed except the terms are added instead of subtracted. An example adds 11/3 + 5/4 and gets 53/60.
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2. Several example problems are worked through, applying the concepts of ratio, proportion, and variation to divide amounts of money and quantities in given ratios.
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- Graphing quadratic functions in standard and vertex forms
- Identifying minimums, maximums, and axes of symmetry from graphs
- Finding minimums and maximums without graphing using calculators
- Applications of quadratic functions like quadratic regression and maximizing enclosed areas with fencing
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This document discusses quadratic functions in vertex form. It defines quadratic functions as having an x^2 term as the highest power of x. Vertex form is defined as y = a(x - h)^2 + k, where (h, k) is the vertex. The document shows how to graph functions in vertex form by sketching the parent graph y = x^2 and then performing transformations based on the a, h, and k values in the function. Several examples are worked through to demonstrate finding the vertex and axis of symmetry.
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The document provides examples and steps for factoring different types of polynomials:
1) It shows how to factor binomials that are the difference of squares, like x^2 - 9, by finding the roots.
2) It demonstrates factoring a perfect square trinomial by identifying the pattern a^2 + 2ab + b^2 and grouping terms.
3) It presents the formula for factoring the sum or difference of two cubes, like 8x^3 + 27, by treating it as (a)^3 + (b)^3 and grouping the terms.
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The document discusses quadratic functions and their graphs. It explores expressing quadratic functions in different forms, and transforming quadratic functions and their graphs through translations, stretches, and other changes. Key aspects covered include the vertex and line of symmetry of parabolas, and how changing the coefficients affects the graph shape and position. Examples of specific quadratic functions are analyzed and graphed.
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This document discusses quadratic equations and their properties. It defines quadratic equations as equations of the form y=ax^2 +bx + c, where the highest power is 2. It explains that quadratic equations can be solved using the quadratic formula, x = -b ± √(b^2 - 4ac) / 2a. The number of solutions depends on the discriminant, b^2 - 4ac. If it is greater than 0, there are two solutions, if equal to 0 there is one solution, and if less than 0 there are no solutions. Examples are provided to demonstrate solving quadratic equations.
Alg March 24, 2009
The document is solving the equation 4(2x) = 2y to determine x in terms of y. It shows that by solving the equation for x, the solution is x = y - 2.
Rational Graphing June 1 2009
This document provides steps for graphing rational functions:
1) Find and sketch any vertical asymptotes where the denominator is zero.
2) Find and sketch the horizontal asymptote based on the degrees of the numerator and denominator.
3) Find the y-intercept by evaluating the function at x=0.
4) Find any x-intercepts by setting the numerator equal to zero.
5) Use sign analysis to determine where the function is positive or negative.
6) Use smooth curves to complete the graph.
Alg May 13, 2009
This document contains math homework assignments and practice problems involving algebra concepts like solving equations, adding and subtracting fractions, working with rational expressions, and factoring polynomials. Students are assigned worksheet problems 4, 7, 10, and 14-16 from section 12-6 for homework. There will be a quiz tomorrow on multiplying, dividing, adding, and subtracting. The document then provides 4 sample word problems for students to practice these skills on their own.
Probability 01.14.08
The document contains 5 questions about probability. Question 1 asks for the probability of getting exactly 2 green dresses if 6 dresses are sold from a rack containing 15 dresses with different colors. Question 2 asks for the probability of a couple having at least 2 girls if they intend to have 4 children. Question 3 asks for the probability that a randomly chosen vehicle from a car lot is an SUV given that the vehicle has air conditioning. Question 4 asks for the probability that exactly one of two randomly drawn razor blades from a box is defective. Question 5 asks for the probability of getting exactly 2 heads if a biased coin is tossed 6 times with the probability of heads being 2/5.
quadraticequations-111211090004-phpapp02 (2).pdf
Quadratic equations take the form of ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to 0. There are three main ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring involves finding two linear expressions whose product is the quadratic expression. Completing the square transforms the equation into the form (x + p)^2 = q. The quadratic formula provides exact solutions for x in terms of a, b, and c. The discriminant, b^2 - 4ac, determines the nature of the roots.
DEV
The document provides step-by-step worked solutions to pre-calculus problems involving sketching graphs based on transformations of an original function. The first problem involves calculating the length of an arc, area of a sector, and angle of an arc given information about a crop circle. The second problem involves sketching the graphs of three transformations - a stretch, inverse, and reciprocal - of an original function given in a graph. Detailed explanations and mathematical working is shown for arriving at the solutions to each part of the two problems.
Math1 1
The document discusses several fundamental concepts of algebra including:
1. Different types of numbers such as integers, rational numbers, and irrational numbers.
2. Properties of operations like addition, subtraction, multiplication, and division.
3. Exponent rules for simplifying expressions with exponents like multiplying terms with the same base.
DEV
The document provides worked solutions to pre-calculus problems involving functions, graphs, and coordinate geometry. It includes step-by-step explanations for determining the length of an arc, the area of a sector of a circle, and the measure of an angle based on an arc length. It also shows how to sketch the graphs of a stretched, compressed, and inverted function based on a given graph. Detailed calculations and graphs are presented to demonstrate the solutions.
Developing Expert Voices
The document provides worked solutions to pre-calculus problems involving functions, graphs, and coordinate transformations. It includes step-by-step explanations for determining the length of an arc, the area of a sector, and the measure of an angle based on given information about a circle. It also demonstrates how to sketch the graphs of transformed functions by applying stretches, translations, and inversions to the coordinates of an original graph.
quadraticequations-111211090004-phpapp02 (1).pdf
The document defines quadratic equations as polynomial equations of degree 2 in the form of ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. It provides examples of quadratic equations in standard and "hidden" forms. Methods for solving quadratics include factoring, completing the square, and using the quadratic formula. The discriminant is used to determine the number and type of roots.
AA Section 8-7
The document discusses powers and roots of negative numbers. Some key points covered include:
- When n is even, n√x is positive for any real number x. When n is odd, n√x is positive if x is positive and negative if x is negative.
- For negative numbers x and y, x^y = xy.
- We can take nth roots of negative numbers when n is an odd integer and x is negative. In this case, the real nth root of x is -x^(1/n).
- We cannot take even roots of negative numbers.
It provides examples of simplifying expressions with negative powers and roots.
Alg April 1, 2009
The document presents a multi-step math problem where a number n is:
1) Increased by 5 and multiplied by 5
2) The result is decreased by 5
3) Finally, the result is divided by 5
The final result is n + 4.
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Alg April 2, 2009
- 1. April 2, 2009 If half of a number is equal to 4 more than twice the number, what is the number? (A) 3 (B) 8/3 (C) 0 (D) 7/2 (E) 4
- 2. Approximating Square Root Values 1 2 3 4 5 6 What about negatives? 7 1 8 9 10
- 3. Vertex (max. or min.) c value b x = 2a (axis of symmetry) Plug in xvalue to find y x intercepts a value Set y = 0 Solve for x
- 4. Graphing Overview Look at avalue is it a or a b Find vertex x = 2a Plug in x to find y Either: Find 2 points to the right of vertex Or: Find zeros
- 5. You Hate Factoring? Well, One More Way to Solve Quadratics! b ± √b2 4ac Quadratic Formula: x = 2a Can be used for ANY quadratic equation Quadratic must be in the form: 0 = ax2 + bx + c y2 = 39y 17 Example:
- 6. HW: Practice 106 Wksht #226 (1/3), 53, 55 Just find a, b, and c; set up word problems Practice 103 Wksht #3651 (1/3)